Curious matrix, how can I easiest explain it's behaviour?

Question

I just found this matrix

$${\bf M} = \left[\begin{array}{cc}1&0.5\\-8&-3\end{array}\right]$$

I have not figured out if my factorization makes sense:

$${\bf M} = \left[\begin{array}{cc}1&0\\-8&1\end{array}\right] \left[\begin{array}{cc}1&\frac{1}{2}\\0&1\end{array}\right]$$

The curious behaviour I want to explain is:

$$({{\bf M}^k})_{11} = (-1)^{k-1}(2k-1)$$

Does the factorization help in explaining the behavior or would we do better trying something else?

Plot over value of element as function of exponent.

Answer 1

Following @LordSharktheUnknown 's comment. Just use Jordan-decomposition. Observe \begin{align} M = \begin{bmatrix} -\frac{1}{4} & -\frac{1}{8}\\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & 1\\ 0 & -1 \end{bmatrix} \begin{bmatrix} 0 & 1\\ -8 & -2 \end{bmatrix} \end{align} which means \begin{align} M^k = \begin{bmatrix} -\frac{1}{4} & -\frac{1}{8}\\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & 1\\ 0 & -1 \end{bmatrix}^k \begin{bmatrix} 0 & 1\\ -8 & -2 \end{bmatrix} = \begin{bmatrix} -\frac{1}{4} & -\frac{1}{8}\\ 1 & 0 \end{bmatrix} \begin{bmatrix} (-1)^k & (-1)^{k-1}k\\ 0 & (-1)^k \end{bmatrix} \begin{bmatrix} 0 & 1\\ -8 & -2 \end{bmatrix}. \end{align} I will leave it to you to multiply everything.